Isogenies between $K3$ surfaces of the Apéry-Fermi pencil
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866916429765279744 |
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| author | Bertin, Marie José Lecacheux, Odile |
| author_facet | Bertin, Marie José Lecacheux, Odile |
| contents | Elliptic fibrations of $K3$ surfaces belonging to the Apéry-Fermi pencil ($Y_k$) may have $2$ or $3$-torsion sections defining on $(Y_k)$ automorphisms $τ$ of order $2$ or $3$. First we consider $Y_{k}/τ$ \ for some fibrations of the singular $K3$ surface $Y_{10}$ in the case of two-torsion sections and obtain as for the singular surface $Y_{2}$ either the Kummer surface associated to $Y_{10}$ or $Y_{10}$ itself. This last case is associated with the complex multiplication on $Y_{10}$. We prove also that for all the fibrations of $Y_{2}$ with $3$-torsion sections $Y_{2}/τ=Y_{10}.$ Results are different for $Y_{10}$ where we can obtain for $Y_{10}/τ$ one of the two surfaces with transcendental lattice $[4 \quad 0\quad 18]$ or $\left[2 \quad 0 \quad 36\right]$. We also explicitly link $3$-isogeny on a fibration and base change on other fibrations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_04151 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Isogenies between $K3$ surfaces of the Apéry-Fermi pencil Bertin, Marie José Lecacheux, Odile Algebraic Geometry 11F23, 11G05, 14J28 (Primary), 14J27 Elliptic fibrations of $K3$ surfaces belonging to the Apéry-Fermi pencil ($Y_k$) may have $2$ or $3$-torsion sections defining on $(Y_k)$ automorphisms $τ$ of order $2$ or $3$. First we consider $Y_{k}/τ$ \ for some fibrations of the singular $K3$ surface $Y_{10}$ in the case of two-torsion sections and obtain as for the singular surface $Y_{2}$ either the Kummer surface associated to $Y_{10}$ or $Y_{10}$ itself. This last case is associated with the complex multiplication on $Y_{10}$. We prove also that for all the fibrations of $Y_{2}$ with $3$-torsion sections $Y_{2}/τ=Y_{10}.$ Results are different for $Y_{10}$ where we can obtain for $Y_{10}/τ$ one of the two surfaces with transcendental lattice $[4 \quad 0\quad 18]$ or $\left[2 \quad 0 \quad 36\right]$. We also explicitly link $3$-isogeny on a fibration and base change on other fibrations. |
| title | Isogenies between $K3$ surfaces of the Apéry-Fermi pencil |
| topic | Algebraic Geometry 11F23, 11G05, 14J28 (Primary), 14J27 |
| url | https://arxiv.org/abs/2203.04151 |